A261174 - OEIS

%I #12 Apr 18 2021 20:41:41

%S 1,2,9,30,90,248,650,1560,3560,7680,15786,31076,58905,107768,191180,

%T 329664,554038,909558,1461655,2302950,3563482,5422392,8124040,

%U 11997648,17482295,25156872,35779092,50330364,70072640,96615760,131999058,178786960,240186182,320179470

%N Number of multigraphs on 4 unlabeled nodes with n edges where the edges can be of two colors.

%H Andrew Howroyd, <a href="/A261174/b261174.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1 - 2*x + 5*x^2 + 2*x^3 + 10*x^4 + 12*x^5 + 32*x^6 + 20*x^7 + 56*x^8 + 20*x^9 + 32*x^10 + 12*x^11 + 10*x^12 + 2*x^13 + 5*x^14 - 2*x^15 + x^16)/((1 - x)^12*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^4). - _Andrew Howroyd_, Apr 18 2021

%t Needs["Combinatorica`"];n = 4; nn = 25; CoefficientList[Series[PairGroupIndex[SymmetricGroup[n], s] /.Table[s[i] -> 1/(1 - x^i)^2, {i, 1, Binomial[n, 2]}], {x, 0, nn}], x]

%o (PARI)

%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

%o edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}

%o G(n)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p, i->(1-x^i)^2)); s/n!}

%o { Vec(G(4) + O(x^36)) } \\ _Andrew Howroyd_, Apr 18 2021

%Y Cf. A050531 (case of 3 nodes).

%K nonn

%O 0,2

%A _Geoffrey Critzer_, Aug 10 2015

%E Terms a(26) and beyond from _Andrew Howroyd_, Apr 18 2021